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Properties

Label 324.6.e
Level $324$
Weight $6$
Character orbit 324.e
Rep. character $\chi_{324}(109,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $10$
Sturm bound $324$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(324\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(324, [\chi])\).

Total New Old
Modular forms 576 40 536
Cusp forms 504 40 464
Eisenstein series 72 0 72

Trace form

\( 40 q - 145 q^{7} + O(q^{10}) \) \( 40 q - 145 q^{7} + 905 q^{13} + 1478 q^{19} - 9194 q^{25} - 7144 q^{31} + 45818 q^{37} - 16804 q^{43} - 82275 q^{49} - 115272 q^{55} + 50279 q^{61} - 26479 q^{67} + 59630 q^{73} - 258055 q^{79} + 38106 q^{85} + 6322 q^{91} - 92977 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.6.e.a 324.e 9.c $2$ $51.964$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-54\) \(88\) $\mathrm{SU}(2)[C_{3}]$ \(q-54\zeta_{6}q^{5}+(88-88\zeta_{6})q^{7}+(-540+\cdots)q^{11}+\cdots\)
324.6.e.b 324.e 9.c $2$ $51.964$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-236\) $\mathrm{U}(1)[D_{3}]$ \(q+(-236+236\zeta_{6})q^{7}-1202\zeta_{6}q^{13}+\cdots\)
324.6.e.c 324.e 9.c $2$ $51.964$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(25\) $\mathrm{U}(1)[D_{3}]$ \(q+(5^{2}-5^{2}\zeta_{6})q^{7}+427\zeta_{6}q^{13}-1711q^{19}+\cdots\)
324.6.e.d 324.e 9.c $2$ $51.964$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(54\) \(88\) $\mathrm{SU}(2)[C_{3}]$ \(q+54\zeta_{6}q^{5}+(88-88\zeta_{6})q^{7}+(540+\cdots)q^{11}+\cdots\)
324.6.e.e 324.e 9.c $4$ $51.964$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-58\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+(-29+29\beta _{1})q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)
324.6.e.f 324.e 9.c $4$ $51.964$ \(\Q(\sqrt{-3}, \sqrt{41})\) None \(0\) \(0\) \(0\) \(32\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+(2^{4}-2^{4}\beta _{1}+3\beta _{2}+3\beta _{3})q^{7}+\cdots\)
324.6.e.g 324.e 9.c $4$ $51.964$ \(\Q(\sqrt{-3}, \sqrt{41})\) None \(0\) \(0\) \(0\) \(32\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+(2^{4}-2^{4}\beta _{1}-3\beta _{2}-3\beta _{3})q^{7}+\cdots\)
324.6.e.h 324.e 9.c $6$ $51.964$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(-33\) \(30\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-11+\beta _{2}+11\beta _{3}+\beta _{5})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
324.6.e.i 324.e 9.c $6$ $51.964$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(0\) \(33\) \(30\) $\mathrm{SU}(2)[C_{3}]$ \(q+(11\beta _{3}+\beta _{5})q^{5}+(10-\beta _{1}-10\beta _{3}+\cdots)q^{7}+\cdots\)
324.6.e.j 324.e 9.c $8$ $51.964$ 8.0.\(\cdots\).10 None \(0\) \(0\) \(0\) \(-176\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{4}+2\beta _{5})q^{5}+(-44+44\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)